A new set distance and its application to shape registration

We propose a new distance measure, called Complement weighted sum of minimal distances, between finite sets in ${\mathbb Z }^n$ and evaluate its usefulness for shape registration and matching. In this set distance the contribution of each point of each set is weighted according to its distance to the complement of the set. In this way, outliers and noise contribute less to the new similarity measure. We evaluate the performance of the new set distance for registration of shapes in binary images and compare it to a number of often used set distances found in the literature. The most extensive evaluation uses a set of synthetic 2D images. We also show three examples of real problems: registering a set of 2D images extracted from synchrotron radiation micro-computed tomography (SR $\upmu $ CT) volumes depicting bone implants; the difficult multi-modal registration task of finding the exact location of a 2D slice of a bone implant, as imaged by a light microscope, within a 3D SR $\upmu $ CT volume of the same implant; and finally recognition of handwritten characters. The evaluation shows that our new set distance performs well for all tasks and outperforms the other observed distance measures in most cases. It is therefore useful in many image registration and shape comparison tasks.